Because of the high absorption localized in a geometrically small zone, the diffusion approximation is not suited to treat properly the neutron distribution in the vicinity of a control rod. The use of transport theory is required, but a full space-dependent transport calculation over the complete core is not usually within the capability of the available computers, and not even necessary, so that various simplifications are needed.

The control-rod cell must be in any case calculated with a transport theory code, whereas the whole reactor is calculated with few-group diffusion theory codes. Two possibilities are available. The control rod can either be simulated by a “non-diffusion region” at whose boundary is given an extrapolation length obtained from the transport calculation, or the control cell can be poisoned with a diffused poison giving the same neutron absorption as the control rod.

The first type of approximation may lead to better results, especially in the flux dis­tribution, but requires a high number of mesh points to treat with sufficient accuracy the flux gradient near the control rods. In this case the cross-section of the control rod has to be simulated using the mesh points of R-9 or X-Y geometry; it is important to have the same absorbing surface as in the real rod, trying not to distort the shape too much.

The problems posed by the calculation of the extrapolation length for “non-diffusion regions” have been dealt with in § 5.5.

The second approximation is used for less detailed calculations (e. g. in the diffusion part of space-dependent burn-up codes), or when the number of control rods is very high. This method becomes rather unreliable when control rods are in the vicinity of zone boundaries (e. g. between core and reflector) or when the rods are few and unsymmetrically distributed.

When hexagonal fuel elements are treated with a diffusion theory code based on hexagonal geometry this method of cell poisoning may be the only way of representing control rods. A transport theory calculation is performed (usually with S„ methods) over the control-rod cell and for each group an absorption cross-section, equivalent to the control rod, is calculated. The control rod is then represented by a uniformly distributed “control poison” whose cross-section as a function of energy is established in such a way that in each group the poison absorbs the same number of neutrons as the control rod. In practice this poison can always be represented by a 1/u absorber with appropriate energy-dependent, self-shielding factors. These self-shieldings take into account the fact that while the cross-section of a 1/u absorber continuously increases with decreasing energy, a control rod cannot be more than black, so that a saturation is reached after which the absorption does not increase. Instead of distributing it homogeneously over the cell it is also possible to concentrate the control poison on a smaller region. This solution is used to represent a ring of control rods with a so called “grey curtain”.

In this way the three-dimensional problem representing a ring of equally spaced control rods can be treated in a two-dimensional RZ geometry. This sort of representa­tion is adequate if a high number of control rods is present in the ring. In this case the concentration of control poison in the curtain must be obtained iterating in a diffusion calculation. The poison concentration is varied until keff is the same as that obtained by a detailed calculation with a more exact representation of the control rods (e. g. an R6 calculation where control rods are calculated as non-diffusion regions).

If more control-rod rings are present, this procedure may have to be repeated for each ring.

When some of the control rods are partially inserted at different depth a three­dimensional code is needed to describe accurately the problem. Codes of this type are now available for big computers, but the high calculation time and sometimes the difficulty of having access to big computers limits the use of three-dimensional codes. On the other hand, three-dimensional diffusion codes with a limited number of mesh points are not giving more information than properly used two-dimensional codes. Fully inserted control rods can be easily calculated in R-в or X-Y geometry.

Partially inserted rods can be calculated in RZ geometry using some approximation. A central control rod can be calculated exactly. If control rods are ordered in rings, so that many rods are on the same radius, they can be simulated with a grey curtain.

Another possibility first proposed by Kalnaes’” consists in the simulation in R-Z geometry of a number of control rods vertically inserted at the same radius with a proper number of horizontal control-rod rings (“piston rings”) of equal radius, whose axial spacing is chosen in such a way that the total length of rod in the reactor remains the same. In this case partial insertions can be calculated. All these methods are valid if a sufficient number of rods is inserted in the core.

It is also important to treat properly the layers of structural material, or voidage around the control rods. This is usually done in the transport calculation and included in the extrapolation length used in the diffusion codes (which must be given for the outermost boundary of all the above-mentioned layers). A limited empty volume around a control rod can increase its effectiveness because the neutron density in the gap tends to be more constant than if the gap were filled with core material, thus reducing the “bottleneck” effect (see ref. 3, p. 626). This name is given to the reduction in absorption in cylindrical geometry as compared with plane geometry due to the shrinking of the area available to the neutrons as they diffuse towards the absorber.

Depending on the moderation ratio, also graphite layers can increase the control-rod effectiveness by increasing the thermal flux in its vicinity.

The control-rod worth depends also on the diffusion length of the surrounding medium, and therefore on the fuel loading. A higher diffusion length increases the effectiveness by increasing the area influenced by the control rod. In the calculation of the control-rod holes when the rods are extracted. Filling the holes with material as is usual with diffusion codes leads to an overestimate of the control-rod worth, which has to be assessed (e. g. Behrens and Benoist methods, §8.10). to be assessed (e. g. Behrens and Benoist methods).

Like the reactivity requirement Дkr the control-rod effectiveness Дke is also calculated as a difference of two static calculations, one without rods and one with fully inserted rods, omitting those which are not supposed to be available or which fail to enter the core.

The determination of Дke (and in general of all reactivity differences) requires great consistency in the two calculations out of which the difference is made. The type of approximation and the spatial meshes must be the same, otherwise numerical differ­ences could alter the result.

The control-rod efficiency calculations should be performed for the cold unpoisoned reactor because this is the condition posing the most stringent requirements on the control system. Both calculations of Дkr and Д/с* involve an uncertainty which may be

assumed to be of the order of 5-10%. The shut-down margin Afce-Afcr must be wide enough to cover this uncertainty.